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Relatively Hyperbolic Groups RELATIVELY HYPERBOLIC GROUPS B. H. BOWDITCH Abstract. In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly dis- continuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be “fine” if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometri- cally finite action of the group is equivariantly homeomorphic to this boundary. This generalises a result of Tukia for geometri- cally finite kleinian groups. We also describe when the boundary is connected. 1. Introduction The aim of this paper is to give an account of the notion of a “rel- atively hyperbolic group”, or more precisely, a group which is “hy- perbolic relative to” a preferred class of “peripheral subgroups”. This notion was introduced by Gromov in his original paper on hyperbolic groups, [Grom]. Since then, it has been a major feature of geomet- ric group theory, with many naturally occurring groups falling into this class. It simultaneously generalises the idea of a hyperbolic group (the case where there are no peripheral subgroups) and that of a ge- ometrically finite kleinian group (where the peripheral subgroups are the maximal parabolic subgroups). Elaborations on the notion of rel- ative hyperbolicity were given by Farb [Fa] and Szczepa´nski [Sz]. In this paper, we describe some alternative viewpoints. Since the first draft of this paper was written there have been numerous papers mak- ing use of relative hyperbolicity, and one could compile an extensive Date: September 2011. 1 2 B. H. BOWDITCH bibliography. Here, however, we will confine ourselves to discussing equivalent formulations of the notion. In that regard, we mention [Y, Da, Bu, DrS, O, GrovM, Gera]. We will say more about these later in the introduction. Of course, these papers include many other references to the subject. We shall give two equivalent definitions of hyperbolicity for a group, Γ, relative to a set, G, of infinite subgroups. Definition 1. We say that Γ is hyperbolic relative to G, if Γ admits a properly discontinous isometric action on a path-metric space, X, with the following properties. (1) X is proper (i.e. complete and locally compact) and hyperbolic, (2) every point of the boundary of X is either a conical limit point or a bounded parabolic point, (3) the elements of G are precisely the maximal parabolic subgroups of Γ, and (4) every element of G is finitely generated. Definition 2. We say that Γ is hyperbolic relative to G, if Γ admits an action on a connected graph, K, with the following properties: (1) K is hyperbolic, and each edge of K is contained in only finitely many circuits of length n for any given integer, n, (2) there are finitely many Γ-orbits of edges, and each edge stabiliser is finite, (3) the elements of G are precisely the infinite vertex stabilisers of K, and (4) every element of G is finitely generated. The equivalence of these definitions will be proven in this paper (see Theorem 7.10). We refer to the elements of G as “peripheral subgroups”. The insistence that they be finitely generated is rather artificial, though it seems to be needed for certain results. The first definition could be viewed as a dynamical characterisation, except that we need to assume that we are dealing with the bound- ary of a proper hyperbolic space, and that the action arises from an isometric action on that space. In fact, it has been shown in [Y] that these last assumptions can be dispensed with. One therefore obtains a purely dynamical characterisation of a relatively hyperbolic group as a convergence group acting on a compact metrisable space, for which every point is either a conical limit point or a bounded parabolic point. This generalises the result of [Bo6], where it was assumed that there are no parabolics. (The argument of [Y] makes use of the fact, proven in [T3], that there are only finitely many conjugacy classes of maximal parabolic subgroups.) RELATIVELY HYPERBOLIC GROUPS 3 The second definition (which we shall study first in this paper) will be phrased formally in terms of group actions on sets, which are “cofi- nite” (i.e. finitely many orbits). We keep much of the discussion fairly general. In principle, one could take a similar approach to studying other “geometric” properties of groups relative to a given set of sub- groups. The standard (non-relative) case corresponds to those actions where all point stabilisers are finite. In this case, the interesting geo- metric properties seem to be quasiisometry invariant, though, in the more general setting, there are natural geometric properties which are not quasiisometry invariant, such as the property we call “fineness” in Section 4. This approach is extended to group splittings in [Bo9]. Each of the various formulations of relative hyperbolicity have tech- nical advantages in specific contexts, and each make apparent certain features not apparent from the others. We aim to explore such matters in this paper. There is also a weaker notion, which we shall refer to here as “weak relative hyperbolicity”. This is a weakening of Definition 2 above. Let G be a conjugacy invariant collection of infinite subgroups of a group Γ. Definition. We say that Γ is weakly hyperbolic relative to G if it admits an action on a connected graph, K, with the following properties: (1) K is hyperbolic, (2) there are finitely many Γ-orbits of edges, and (3) each element of G fixes a vertex of K, and each vertex stabiliser of K contains an element of G as a subgroup of finite index. This gives rise to many more examples (see [Fa, Gers, Mas]). How- ever, there seems to be a greater degree of arbitrariness about this definition. In this paper, we shall focus on the stronger definition. We briefly elaborate on these definitions, beginning with the dynamical formulation. Suppose that a group, Γ, acts by homeomorphism on a perfect metrisable compactum, M. We say that Γ is a “convergence group” (in the sense of [GehM]) if the induced action on the space of distinct triples is properly discontinuous. (For further discussion, see [GehM, T2, Fr1, Bo7].) We say that a convergence group is “(minimal) geometrically finite” if every point of M is either a conical limit point or a bounded parabolic point. (For definitions, see Section 6.) This is a natural generalisation of the Beardon-Maskit definition [BeM]. It has also been considered in this generality in [T3] and [Fr2]. It generalises the classical formulation of geometrical finiteness described by Ahlfors and Greenberg [A, Gre], see [Bo2, Bo4]. 4 B. H. BOWDITCH In this paper, we shall mostly confine ourselves to the case where M is the boundary, ∂X, of a complete, locally compact, Gromov hyperbolic space, X. If Γ acts properly discontinuously and isometrically on X, then the induced action of ∂X is a convergence action. We can thus define an action on X to be “geometrically finite” if the induced action on ∂X is geometrically finite in the dynamical sense. If Π ⊆ ∂X is the set of parabolic points, then one can show that Π/Γ is finite. In fact, one can construct a strictly invariant system of horoballs, (B(p))p∈Π, about Π, such that (X \ Sp∈Π int B(p))/Γ is compact. In this way, we recover a generalisation of Marden’s definition of geometrical finiteness [Mar]. This gives us the first definition of relative hyperbolicity, namely that a group Γ is hyperbolic relative to a set of infinite finitely generated subgroups (the “peripheral subgroups”) if it admits a geometrically finite action on a proper hyperbolic space, where these subgroups are precisely the maximal parabolic subgroups. We note immediately that the intersection of any two peripheral subgroups is finite, and that there are finitely many conjugacy classes of peripheral subgroups. Moreover, each peripheral subgroup is equal to its normaliser. Note that, in the case of a hyperbolic group (where G = ∅) the quasiisometry class of the space X depends only on the group, Γ. This allows us to define the boundary, denoted ∂Γ, of Γ as ∂X. In the gen- eral situation, however, this is not the case. Indeed, there seems to be no particularly natural choice of quasiisometry class of space X. How- ever, as we shall discuss in a moment, we can associate to a relatively hyperbolic group, (Γ, G), a space X(Γ), together with a geometrically finite action of Γ, which is canonical up to Γ-equivariant quasiisometry, even though the construction is somewhat artificial. This allows us to give a formal definition of the boundary, ∂(Γ, G) as ∂X(Γ). One can go on to show that if X is any complete locally compact hyperbolic space admitting a geometrically finite action of Γ, then the limit set of ∂X is, in fact, Γ-equivariantly homeomorphic to ∂(Γ, G). We thus see that ∂(Γ, G) is a quite natural object to associate to (Γ, G), even if X(Γ) is not.
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